Complete solving of explicit evaluation of Gauss sums in the index 2 case

By Jing Yang, Lingli Xia. 2009

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AbstractLet p be a prime number, N be a positive integer such that gcd(N, p) = 1, q = pf where f is the multiplicative order of p modulo N. Let χ be a primitive multiplicative character of order N over finite field $$ \mathbb{F}_q $$. This paper studies the problem of explicit evaluation of Gauss sums G(χ) in the “index 2 case” (i.e. [(ℤ/Nℤ)*: 〈p〉] = 2). Firstly, the classification of the Gauss sums in the index 2 case is presented. Then, the explicit evaluation of Gauss sums G(χλ) (1 ⩽ λ ⩽ N − 1) in the index 2 case with order N being general even integer (i.e. N = 2r · N0, where r, N0 are positive integers and N0 ⩾ 3 is odd) is obtained. Thus, combining with the researches before, the problem of explicit evaluation of Gauss sums in the index 2 case is completely solved.

Subfields: Algebra and Number Theory, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dd11e60f2fb35206a09656fb5e8fa8858e0fd1dc